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Rishabh Iyer

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Published work

27 published item(s)

preprint2026arXiv

SMA: Submodular Modality Aligner For Data Efficient Multimodal Learning

Despite the recent success of Multimodal Foundation Models (FMs), their reliance on massive paired datasets limits their applicability in low-data and rare-scenario settings where aligned data is scarce and expensive. A key bottleneck is the adoption of an instance-level formulation, which learns alignment by maximizing correlation between individual image-text pairs while neglecting the underlying geometric structure across modalities resulting in a modality gap across input modalities. In this paper, we propose a combinatorial paradigm for multimodal alignment that moves beyond pairwise learning and introduce the \emph{Submodular Modality Aligner (SMA)}, which treats multiple augmentations and descriptions of an entity as a set, leveraging multiple descriptions of the data to capture richer cross-modal structure. We instantiate SMA using a principled objective based on Submodular Mutual Information (SMI), which jointly maximizes inter-modality mutual information while reducing cross-modal divergence. This formulation enables the model to effectively utilize multiple positive associations and extract significantly more information from limited data. We evaluate SMA on 14 zero-shot classification and retrieval tasks from the CLIP benchmark and demonstrate consistent gains in the low-data regime. Notably, SMA achieves strong multimodal generalization using only tens of thousands of samples. This is orders of magnitude fewer than standard approaches. Our results highlight the importance of set-based formulations and submodular objectives for data-efficient multimodal learning.

preprint2022arXiv

Active Data Discovery: Mining Unknown Data using Submodular Information Measures

Active Learning is a very common yet powerful framework for iteratively and adaptively sampling subsets of the unlabeled sets with a human in the loop with the goal of achieving labeling efficiency. Most real world datasets have imbalance either in classes and slices, and correspondingly, parts of the dataset are rare. As a result, there has been a lot of work in designing active learning approaches for mining these rare data instances. Most approaches assume access to a seed set of instances which contain these rare data instances. However, in the event of more extreme rareness, it is reasonable to assume that these rare data instances (either classes or slices) may not even be present in the seed labeled set, and a critical need for the active learning paradigm is to efficiently discover these rare data instances. In this work, we provide an active data discovery framework which can mine unknown data slices and classes efficiently using the submodular conditional gain and submodular conditional mutual information functions. We provide a general algorithmic framework which works in a number of scenarios including image classification and object detection and works with both rare classes and rare slices present in the unlabeled set. We show significant accuracy and labeling efficiency gains with our approach compared to existing state-of-the-art active learning approaches for actively discovering these rare classes and slices.

preprint2022arXiv

AUTOMATA: Gradient Based Data Subset Selection for Compute-Efficient Hyper-parameter Tuning

Deep neural networks have seen great success in recent years; however, training a deep model is often challenging as its performance heavily depends on the hyper-parameters used. In addition, finding the optimal hyper-parameter configuration, even with state-of-the-art (SOTA) hyper-parameter optimization (HPO) algorithms, can be time-consuming, requiring multiple training runs over the entire dataset for different possible sets of hyper-parameters. Our central insight is that using an informative subset of the dataset for model training runs involved in hyper-parameter optimization, allows us to find the optimal hyper-parameter configuration significantly faster. In this work, we propose AUTOMATA, a gradient-based subset selection framework for hyper-parameter tuning. We empirically evaluate the effectiveness of AUTOMATA in hyper-parameter tuning through several experiments on real-world datasets in the text, vision, and tabular domains. Our experiments show that using gradient-based data subsets for hyper-parameter tuning achieves significantly faster turnaround times and speedups of 3$\times$-30$\times$ while achieving comparable performance to the hyper-parameters found using the entire dataset.

preprint2022arXiv

BASIL: Balanced Active Semi-supervised Learning for Class Imbalanced Datasets

Current semi-supervised learning (SSL) methods assume a balance between the number of data points available for each class in both the labeled and the unlabeled data sets. However, there naturally exists a class imbalance in most real-world datasets. It is known that training models on such imbalanced datasets leads to biased models, which in turn lead to biased predictions towards the more frequent classes. This issue is further pronounced in SSL methods, as they would use this biased model to obtain psuedo-labels (on the unlabeled data) during training. In this paper, we tackle this problem by attempting to select a balanced labeled dataset for SSL that would result in an unbiased model. Unfortunately, acquiring a balanced labeled dataset from a class imbalanced distribution in one shot is challenging. We propose BASIL (Balanced Active Semi-supervIsed Learning), a novel algorithm that optimizes the submodular mutual information (SMI) functions in a per-class fashion to gradually select a balanced dataset in an active learning loop. Importantly, our technique can be efficiently used to improve the performance of any SSL method. Our experiments on Path-MNIST and Organ-MNIST medical datasets for a wide array of SSL methods show the effectiveness of Basil. Furthermore, we observe that Basil outperforms the state-of-the-art diversity and uncertainty based active learning methods since the SMI functions select a more balanced dataset.

preprint2022arXiv

Fair Machine Learning under Limited Demographically Labeled Data

Research has shown that, machine learning models might inherit and propagate undesired social biases encoded in the data. To address this problem, fair training algorithms are developed. However, most algorithms assume we know demographic/sensitive data features such as gender and race. This assumption falls short in scenarios where collecting demographic information is not feasible due to privacy concerns, and data protection policies. A recent line of work develops fair training methods that can function without any demographic feature on the data, that are collectively referred as Rawlsian methods. Yet, we show in experiments that, Rawlsian methods tend to exhibit relatively high bias. Given this, we look at the middle ground between the previous approaches, and consider a setting where we know the demographic attributes for only a small subset of our data. In such a setting, we design fair training algorithms which exhibit both good utility, and low bias. In particular, we show that our techniques can train models to significantly outperform Rawlsian approaches even when 0.1% of demographic attributes are available in the training data. Furthermore, our main algorithm can accommodate multiple training objectives easily. We expand our main algorithm to achieve robustness to label noise in addition to fairness in the limited demographics setting to highlight that property as well.

preprint2022arXiv

GCR: Gradient Coreset Based Replay Buffer Selection For Continual Learning

Continual learning (CL) aims to develop techniques by which a single model adapts to an increasing number of tasks encountered sequentially, thereby potentially leveraging learnings across tasks in a resource-efficient manner. A major challenge for CL systems is catastrophic forgetting, where earlier tasks are forgotten while learning a new task. To address this, replay-based CL approaches maintain and repeatedly retrain on a small buffer of data selected across encountered tasks. We propose Gradient Coreset Replay (GCR), a novel strategy for replay buffer selection and update using a carefully designed optimization criterion. Specifically, we select and maintain a "coreset" that closely approximates the gradient of all the data seen so far with respect to current model parameters, and discuss key strategies needed for its effective application to the continual learning setting. We show significant gains (2%-4% absolute) over the state-of-the-art in the well-studied offline continual learning setting. Our findings also effectively transfer to online / streaming CL settings, showing upto 5% gains over existing approaches. Finally, we demonstrate the value of supervised contrastive loss for continual learning, which yields a cumulative gain of up to 5% accuracy when combined with our subset selection strategy.

preprint2022arXiv

Learning to Robustly Aggregate Labeling Functions for Semi-supervised Data Programming

A critical bottleneck in supervised machine learning is the need for large amounts of labeled data which is expensive and time consuming to obtain. However, it has been shown that a small amount of labeled data, while insufficient to re-train a model, can be effectively used to generate human-interpretable labeling functions (LFs). These LFs, in turn, have been used to generate a large amount of additional noisy labeled data, in a paradigm that is now commonly referred to as data programming. However, previous approaches to automatically generate LFs make no attempt to further use the given labeled data for model training, thus giving up opportunities for improved performance. Moreover, since the LFs are generated from a relatively small labeled dataset, they are prone to being noisy, and naively aggregating these LFs can lead to very poor performance in practice. In this work, we propose an LF based reweighting framework \ouralgo{} to solve these two critical limitations. Our algorithm learns a joint model on the (same) labeled dataset used for LF induction along with any unlabeled data in a semi-supervised manner, and more critically, reweighs each LF according to its goodness, influencing its contribution to the semi-supervised loss using a robust bi-level optimization algorithm. We show that our algorithm significantly outperforms prior approaches on several text classification datasets.

preprint2022arXiv

PLATINUM: Semi-Supervised Model Agnostic Meta-Learning using Submodular Mutual Information

Few-shot classification (FSC) requires training models using a few (typically one to five) data points per class. Meta learning has proven to be able to learn a parametrized model for FSC by training on various other classification tasks. In this work, we propose PLATINUM (semi-suPervised modeL Agnostic meTa-learnIng usiNg sUbmodular Mutual information), a novel semi-supervised model agnostic meta-learning framework that uses the submodular mutual information (SMI) functions to boost the performance of FSC. PLATINUM leverages unlabeled data in the inner and outer loop using SMI functions during meta-training and obtains richer meta-learned parameterizations for meta-test. We study the performance of PLATINUM in two scenarios - 1) where the unlabeled data points belong to the same set of classes as the labeled set of a certain episode, and 2) where there exist out-of-distribution classes that do not belong to the labeled set. We evaluate our method on various settings on the miniImageNet, tieredImageNet and Fewshot-CIFAR100 datasets. Our experiments show that PLATINUM outperforms MAML and semi-supervised approaches like pseduo-labeling for semi-supervised FSC, especially for small ratio of labeled examples per class.

preprint2022arXiv

PRISM: A Rich Class of Parameterized Submodular Information Measures for Guided Subset Selection

With ever-increasing dataset sizes, subset selection techniques are becoming increasingly important for a plethora of tasks. It is often necessary to guide the subset selection to achieve certain desiderata, which includes focusing or targeting certain data points, while avoiding others. Examples of such problems include: i)targeted learning, where the goal is to find subsets with rare classes or rare attributes on which the model is underperforming, and ii)guided summarization, where data (e.g., image collection, text, document or video) is summarized for quicker human consumption with specific additional user intent. Motivated by such applications, we present PRISM, a rich class of PaRameterIzed Submodular information Measures. Through novel functions and their parameterizations, PRISM offers a variety of modeling capabilities that enable a trade-off between desired qualities of a subset like diversity or representation and similarity/dissimilarity with a set of data points. We demonstrate how PRISM can be applied to the two real-world problems mentioned above, which require guided subset selection. In doing so, we show that PRISM interestingly generalizes some past work, therein reinforcing its broad utility. Through extensive experiments on diverse datasets, we demonstrate the superiority of PRISM over the state-of-the-art in targeted learning and in guided image-collection summarization

preprint2022arXiv

Submodlib: A Submodular Optimization Library

Submodular functions are a special class of set functions which naturally model the notion of representativeness, diversity, coverage etc. and have been shown to be computationally very efficient. A lot of past work has applied submodular optimization to find optimal subsets in various contexts. Some examples include data summarization for efficient human consumption, finding effective smaller subsets of training data to reduce the model development time (training, hyper parameter tuning), finding effective subsets of unlabeled data to reduce the labeling costs, etc. A recent work has also leveraged submodular functions to propose submodular information measures which have been found to be very useful in solving the problems of guided subset selection and guided summarization. In this work, we present Submodlib which is an open-source, easy-to-use, efficient and scalable Python library for submodular optimization with a C++ optimization engine. Submodlib finds its application in summarization, data subset selection, hyper parameter tuning, efficient training and more. Through a rich API, it offers a great deal of flexibility in the way it can be used. Source of Submodlib is available at https://github.com/decile-team/submodlib.

preprint2022arXiv

TALISMAN: Targeted Active Learning for Object Detection with Rare Classes and Slices using Submodular Mutual Information

Deep neural networks based object detectors have shown great success in a variety of domains like autonomous vehicles, biomedical imaging, etc. It is known that their success depends on a large amount of data from the domain of interest. While deep models often perform well in terms of overall accuracy, they often struggle in performance on rare yet critical data slices. For example, data slices like "motorcycle at night" or "bicycle at night" are often rare but very critical slices for self-driving applications and false negatives on such rare slices could result in ill-fated failures and accidents. Active learning (AL) is a well-known paradigm to incrementally and adaptively build training datasets with a human in the loop. However, current AL based acquisition functions are not well-equipped to tackle real-world datasets with rare slices, since they are based on uncertainty scores or global descriptors of the image. We propose TALISMAN, a novel framework for Targeted Active Learning or object detectIon with rare slices using Submodular MutuAl iNformation. Our method uses the submodular mutual information functions instantiated using features of the region of interest (RoI) to efficiently target and acquire data points with rare slices. We evaluate our framework on the standard PASCAL VOC07+12 and BDD100K, a real-world self-driving dataset. We observe that TALISMAN outperforms other methods by in terms of average precision on rare slices, and in terms of mAP.

preprint2021arXiv

How Good is a Video Summary? A New Benchmarking Dataset and Evaluation Framework Towards Realistic Video Summarization

Automatic video summarization is still an unsolved problem due to several challenges. The currently available datasets either have very short videos or have few long videos of only a particular type. We introduce a new benchmarking video dataset called VISIOCITY (VIdeo SummarIzatiOn based on Continuity, Intent and DiversiTY) which comprises of longer videos across six different categories with dense concept annotations capable of supporting different flavors of video summarization and other vision problems. For long videos, human reference summaries necessary for supervised video summarization techniques are difficult to obtain. We explore strategies to automatically generate multiple reference summaries from indirect ground truth present in VISIOCITY. We show that these summaries are at par with human summaries. We also present a study of different desired characteristics of a good summary and demonstrate how it is normal to have two good summaries with different characteristics. Thus we argue that evaluating a summary against one or more human summaries and using a single measure has its shortcomings. We propose an evaluation framework for better quantitative assessment of summary quality which is closer to human judgment. Lastly, we present insights into how a model can be enhanced to yield better summaries. Sepcifically, when multiple diverse ground truth summaries can exist, learning from them individually and using a combination of loss functions measuring different characteristics is better than learning from a single combined (oracle) ground truth summary using a single loss function. We demonstrate the effectiveness of doing so as compared to some of the representative state of the art techniques tested on VISIOCITY. We release VISIOCITY as a benchmarking dataset and invite researchers to test the effectiveness of their video summarization algorithms on VISIOCITY.

preprint2021arXiv

Submodular Combinatorial Information Measures with Applications in Machine Learning

Information-theoretic quantities like entropy and mutual information have found numerous uses in machine learning. It is well known that there is a strong connection between these entropic quantities and submodularity since entropy over a set of random variables is submodular. In this paper, we study combinatorial information measures that generalize independence, (conditional) entropy, (conditional) mutual information, and total correlation defined over sets of (not necessarily random) variables. These measures strictly generalize the corresponding entropic measures since they are all parameterized via submodular functions that themselves strictly generalize entropy. Critically, we show that, unlike entropic mutual information in general, the submodular mutual information is actually submodular in one argument, holding the other fixed, for a large class of submodular functions whose third-order partial derivatives satisfy a non-negativity property. This turns out to include a number of practically useful cases such as the facility location and set-cover functions. We study specific instantiations of the submodular information measures on these, as well as the probabilistic coverage, graph-cut, and saturated coverage functions, and see that they all have mathematically intuitive and practically useful expressions. Regarding applications, we connect the maximization of submodular (conditional) mutual information to problems such as mutual-information-based, query-based, and privacy-preserving summarization -- and we connect optimizing the multi-set submodular mutual information to clustering and robust partitioning.

preprint2020arXiv

Concave Aspects of Submodular Functions

Submodular Functions are a special class of set functions, which generalize several information-theoretic quantities such as entropy and mutual information [1]. Submodular functions have subgradients and subdifferentials [2] and admit polynomial-time algorithms for minimization, both of which are fundamental characteristics of convex functions. Submodular functions also show signs similar to concavity. Submodular function maximization, though NP-hard, admits constant-factor approximation guarantees, and concave functions composed with modular functions are submodular. In this paper, we try to provide a more complete picture of the relationship between submodularity with concavity. We characterize the super-differentials and polyhedra associated with upper bounds and provide optimality conditions for submodular maximization using the-super differentials. This paper is a concise and shorter version of our longer preprint [3].

preprint2020arXiv

Realistic Video Summarization through VISIOCITY: A New Benchmark and Evaluation Framework

Automatic video summarization is still an unsolved problem due to several challenges. We take steps towards making automatic video summarization more realistic by addressing them. Firstly, the currently available datasets either have very short videos or have few long videos of only a particular type. We introduce a new benchmarking dataset VISIOCITY which comprises of longer videos across six different categories with dense concept annotations capable of supporting different flavors of video summarization and can be used for other vision problems. Secondly, for long videos, human reference summaries are difficult to obtain. We present a novel recipe based on pareto optimality to automatically generate multiple reference summaries from indirect ground truth present in VISIOCITY. We show that these summaries are at par with human summaries. Thirdly, we demonstrate that in the presence of multiple ground truth summaries (due to the highly subjective nature of the task), learning from a single combined ground truth summary using a single loss function is not a good idea. We propose a simple recipe VISIOCITY-SUM to enhance an existing model using a combination of losses and demonstrate that it beats the current state of the art techniques when tested on VISIOCITY. We also show that a single measure to evaluate a summary, as is the current typical practice, falls short. We propose a framework for better quantitative assessment of summary quality which is closer to human judgment than a single measure, say F1. We report the performance of a few representative techniques of video summarization on VISIOCITY assessed using various measures and bring out the limitation of the techniques and/or the assessment mechanism in modeling human judgment and demonstrate the effectiveness of our evaluation framework in doing so.

preprint2020arXiv

Robust Submodular Minimization with Applications to Cooperative Modeling

Robust Optimization is becoming increasingly important in machine learning applications. This paper studies the problem of robust submodular minimization subject to combinatorial constraints. Constrained Submodular Minimization arises in several applications such as co-operative cuts in image segmentation, co-operative matchings in image correspondence, etc. Many of these models are defined over clusterings of data points (for example pixels in images), and it is important for these models to be robust to perturbations and uncertainty in the data. While several existing papers have studied robust submodular maximization, ours is the first work to study the minimization version under a broad range of combinatorial constraints including cardinality, knapsack, matroid as well as graph-based constraints such as cuts, paths, matchings, and trees. In each case, we provide scalable approximation algorithms and also study hardness bounds. Finally, we empirically demonstrate the utility of our algorithms on synthetic and real-world datasets.

preprint2016arXiv

Mixed Robust/Average Submodular Partitioning: Fast Algorithms, Guarantees, and Applications to Parallel Machine Learning and Multi-Label Image Segmentation

We study two mixed robust/average-case submodular partitioning problems that we collectively call Submodular Partitioning. These problems generalize both purely robust instances of the problem (namely max-min submodular fair allocation (SFA) and min-max submodular load balancing (SLB) and also generalize average-case instances (that is the submodular welfare problem (SWP) and submodular multiway partition (SMP). While the robust versions have been studied in the theory community, existing work has focused on tight approximation guarantees, and the resultant algorithms are not, in general, scalable to very large real-world applications. This is in contrast to the average case, where most of the algorithms are scalable. In the present paper, we bridge this gap, by proposing several new algorithms (including those based on greedy, majorization-minimization, minorization-maximization, and relaxation algorithms) that not only scale to large sizes but that also achieve theoretical approximation guarantees close to the state-of-the-art, and in some cases achieve new tight bounds. We also provide new scalable algorithms that apply to additive combinations of the robust and average-case extreme objectives. We show that these problems have many applications in machine learning (ML). This includes: 1) data partitioning and load balancing for distributed machine algorithms on parallel machines; 2) data clustering; and 3) multi-label image segmentation with (only) Boolean submodular functions via pixel partitioning. We empirically demonstrate the efficacy of our algorithms on real-world problems involving data partitioning for distributed optimization of standard machine learning objectives (including both convex and deep neural network objectives), and also on purely unsupervised (i.e., no supervised or semi-supervised learning, and no interactive segmentation) image segmentation.

preprint2015arXiv

Polyhedral aspects of Submodularity, Convexity and Concavity

Seminal work by Edmonds and Lovasz shows the strong connection between submodularity and convexity. Submodular functions have tight modular lower bounds, and subdifferentials in a manner akin to convex functions. They also admit poly-time algorithms for minimization and satisfy the Fenchel duality theorem and the Discrete Seperation Theorem, both of which are fundamental characteristics of convex functions. Submodular functions also show signs similar to concavity. Submodular maximization, though NP hard, admits constant factor approximation guarantees. Concave functions composed with modular functions are submodular, and they also satisfy diminishing returns property. This manuscript provides a more complete picture on the relationship between submodularity with convexity and concavity, by extending many of the results connecting submodularity with convexity to the concave aspects of submodularity. We first show the existence of superdifferentials, and efficiently computable tight modular upper bounds of a submodular function. While we show that it is hard to characterize this polyhedron, we obtain inner and outer bounds on the superdifferential along with certain specific and useful supergradients. We then investigate forms of concave extensions of submodular functions and show interesting relationships to submodular maximization. We next show connections between optimality conditions over the superdifferentials and submodular maximization, and show how forms of approximate optimality conditions translate into approximation factors for maximization. We end this paper by studying versions of the discrete seperation theorem and the Fenchel duality theorem when seen from the concave point of view. In every case, we relate our results to the existing results from the convex point of view, thereby improving the analysis of the relationship between submodularity, convexity, and concavity.

preprint2015arXiv

Submodular Hamming Metrics

We show that there is a largely unexplored class of functions (positive polymatroids) that can define proper discrete metrics over pairs of binary vectors and that are fairly tractable to optimize over. By exploiting submodularity, we are able to give hardness results and approximation algorithms for optimizing over such metrics. Additionally, we demonstrate empirically the effectiveness of these metrics and associated algorithms on both a metric minimization task (a form of clustering) and also a metric maximization task (generating diverse k-best lists).

preprint2014arXiv

Algorithms for Approximate Minimization of the Difference Between Submodular Functions, with Applications

We extend the work of Narasimhan and Bilmes [30] for minimizing set functions representable as a dierence between submodular functions. Similar to [30], our new algorithms are guaranteed to monotonically reduce the objective function at every step. We empirically and theoretically show that the per-iteration cost of our algorithms is much less than [30], and our algorithms can be used to efficiently minimize a dierence between submodular functions under various combinatorial constraints, a problem not previously addressed. We provide computational bounds and a hardness result on the multiplicative inapproximability of minimizing the dierence between submodular functions. We show, however, that it is possible to give worst-case additive bounds by providing a polynomial time computable lower-bound on the minima. Finally we show how a number of machine learning problems can be modeled as minimizing the dierence between submodular functions. We experimentally show the validity of our algorithms by testing them on the problem of feature selection with submodular cost features.

preprint2014arXiv

The Lovasz-Bregman Divergence and connections to rank aggregation, clustering, and web ranking

We extend the recently introduced theory of Lovasz-Bregman (LB) divergences (Iyer & Bilmes 2012) in several ways. We show that they represent a distortion between a "score" and an "ordering", thus providing a new view of rank aggregation and order based clustering with interesting connections to web ranking. We show how the LB divergences have a number of properties akin to many permutation based metrics, and in fact have as special cases forms very similar to the Kendall-tau metric. We also show how the LB divergences subsume a number of commonly used ranking measures in information retrieval, like NDCG and AUC. Unlike the traditional permutation based metrics, however, the LB divergence naturally captures a notion of "confidence" in the orderings, thus providing a new representation to applications involving aggregating scores as opposed to just orderings. We show how a number of recently used web ranking models are forms of Lovasz-Bregman rank aggregation and also observe that a natural form of Mallow's model using the LB divergence has been used as conditional ranking models for the "Learning to Rank" problem.

preprint2013arXiv

Algorithms for Approximate Minimization of the Difference Between Submodular Functions, with Applications

We extend the work of Narasimhan and Bilmes [30] for minimizing set functions representable as a difference between submodular functions. Similar to [30], our new algorithms are guaranteed to monotonically reduce the objective function at every step. We empirically and theoretically show that the per-iteration cost of our algorithms is much less than [30], and our algorithms can be used to efficiently minimize a difference between submodular functions under various combinatorial constraints, a problem not previously addressed. We provide computational bounds and a hardness result on the mul- tiplicative inapproximability of minimizing the difference between submodular functions. We show, however, that it is possible to give worst-case additive bounds by providing a polynomial time computable lower-bound on the minima. Finally we show how a number of machine learning problems can be modeled as minimizing the difference between submodular functions. We experimentally show the validity of our algorithms by testing them on the problem of feature selection with submodular cost features.

preprint2013arXiv

Curvature and Optimal Algorithms for Learning and Minimizing Submodular Functions

We investigate three related and important problems connected to machine learning: approximating a submodular function everywhere, learning a submodular function (in a PAC-like setting [53]), and constrained minimization of submodular functions. We show that the complexity of all three problems depends on the 'curvature' of the submodular function, and provide lower and upper bounds that refine and improve previous results [3, 16, 18, 52]. Our proof techniques are fairly generic. We either use a black-box transformation of the function (for approximation and learning), or a transformation of algorithms to use an appropriate surrogate function (for minimization). Curiously, curvature has been known to influence approximations for submodular maximization [7, 55], but its effect on minimization, approximation and learning has hitherto been open. We complete this picture, and also support our theoretical claims by empirical results.

preprint2013arXiv

Fast Semidifferential-based Submodular Function Optimization

We present a practical and powerful new framework for both unconstrained and constrained submodular function optimization based on discrete semidifferentials (sub- and super-differentials). The resulting algorithms, which repeatedly compute and then efficiently optimize submodular semigradients, offer new and generalize many old methods for submodular optimization. Our approach, moreover, takes steps towards providing a unifying paradigm applicable to both submodular min- imization and maximization, problems that historically have been treated quite distinctly. The practicality of our algorithms is important since interest in submodularity, owing to its natural and wide applicability, has recently been in ascendance within machine learning. We analyze theoretical properties of our algorithms for minimization and maximization, and show that many state-of-the-art maximization algorithms are special cases. Lastly, we complement our theoretical analyses with supporting empirical experiments.

preprint2013arXiv

Submodular Optimization with Submodular Cover and Submodular Knapsack Constraints

We investigate two new optimization problems -- minimizing a submodular function subject to a submodular lower bound constraint (submodular cover) and maximizing a submodular function subject to a submodular upper bound constraint (submodular knapsack). We are motivated by a number of real-world applications in machine learning including sensor placement and data subset selection, which require maximizing a certain submodular function (like coverage or diversity) while simultaneously minimizing another (like cooperative cost). These problems are often posed as minimizing the difference between submodular functions [14, 35] which is in the worst case inapproximable. We show, however, that by phrasing these problems as constrained optimization, which is more natural for many applications, we achieve a number of bounded approximation guarantees. We also show that both these problems are closely related and an approximation algorithm solving one can be used to obtain an approximation guarantee for the other. We provide hardness results for both problems thus showing that our approximation factors are tight up to log-factors. Finally, we empirically demonstrate the performance and good scalability properties of our algorithms.

preprint2013arXiv

The Lovasz-Bregman Divergence and connections to rank aggregation, clustering, and web ranking

We extend the recently introduced theory of Lovasz-Bregman (LB) divergences (Iyer & Bilmes, 2012) in several ways. We show that they represent a distortion between a 'score' and an 'ordering', thus providing a new view of rank aggregation and order based clustering with interesting connections to web ranking. We show how the LB divergences have a number of properties akin to many permutation based metrics, and in fact have as special cases forms very similar to the Kendall-$τ$ metric. We also show how the LB divergences subsume a number of commonly used ranking measures in information retrieval, like the NDCG and AUC. Unlike the traditional permutation based metrics, however, the LB divergence naturally captures a notion of "confidence" in the orderings, thus providing a new representation to applications involving aggregating scores as opposed to just orderings. We show how a number of recently used web ranking models are forms of Lovasz-Bregman rank aggregation and also observe that a natural form of Mallow's model using the LB divergence has been used as conditional ranking models for the 'Learning to Rank' problem.

preprint2011arXiv

Estimation of the Embedding Capacity in Pixel-pair based Watermarking Schemes

Estimation of the Embedding capacity is an important problem specifically in reversible multi-pass watermarking and is required for analysis before any image can be watermarked. In this paper, we propose an efficient method for estimating the embedding capacity of a given cover image under multi-pass embedding, without actually embedding the watermark. We demonstrate this for a class of reversible watermarking schemes which operate on a disjoint group of pixels, specifically for pixel pairs. The proposed algorithm iteratively updates the co-occurrence matrix at every stage, to estimate the multi-pass embedding capacity, and is much more efficient vis-a-vis actual watermarking. We also suggest an extremely efficient, pre-computable tree based implementation which is conceptually similar to the co-occurrence based method, but provides the estimates in a single iteration, requiring a complexity akin to that of single pass capacity estimation. We also provide bounds on the embedding capacity. We finally show how our method can be easily used on a number of watermarking algorithms and specifically evaluate the performance of our algorithms on the benchmark watermarking schemes of Tian [11] and Coltuc [6].